Vibration effects and acoustic performance of manufactured products, such as for example, but not limited to, transportation means, has become a very important aspect in the design and development process of new products, not only to improve the comfort of the user of the product, e.g. passengers in a car or in an aircraft, but also to reduce the nuisance to the surroundings, e.g. for habitations close to highways or to an airport. The additional trend of mass customisation furthermore forces engineers to evaluate and design a higher number of transportation means variants on a lower number of platforms. The design decisions therefore are based more and more on virtual prototypes, such that the time-to-market as well as the economical cost can be reduced. The latter results in increase of importance of fast assembly predictions.
Many assembly prediction methods are based on sub-structuring procedures, as these aim at reducing the size and subsequent computational cost of an original discretised model of an assembly or complex system. Using sub-structuring, the system is divided into sub-structures, corresponding sub-structure models are condensed separately and the condensed sub-structure models are coupled back together into a reduced system model. The quality of the performance of a sub-structuring method thereby typically is the balanced result between the accuracy loss due to the sub-structure model condensation and the gain in computational efforts due to the model size reduction. Well known sub-structuring methods for studying acoustic and/or vibration properties of complex systems are e.g. Component Mode Synthesis (CMS) and Statistical Energy Analysis (SEA). Whereas CMS is based on quantities such as force and displacement, SEA is an energy-based analysis method directed on providing expected results for energy-related information. The present invention will be oriented to CMS.
Using CMS, a system typically is modelled as an assembly of component sub-structures, also called sub-systems, that interconnect to each other, whereby the substructures typically first are dealt with independently by transforming the physical degrees of freedom (DOFs) of each sub-structure model into a reduced number of so-called generalised co-ordinates, the columns of the transformation matrix used being referred to as Ritz vectors or component modes. Thereafter, the reduced sub-structure models are coupled together by expressing the equations of compatibility and/or continuity between the sub-structures. The degrees of freedom, i.e. the unknowns, of the reduced system model consist of internal degrees of freedom of the sub-structures and interface degrees of freedom, defined at the interfaces between the sub-structures. Typically information about the behaviour of the degrees of freedom at the interface as well as the internal degrees of freedom needs to be known for each component A well-established and widely used CMS variant is the Craig-Bampton method, which uses normal modes of the components as Ritz vectors, whereby the interface degrees of freedom (DOFs) are assumed to be fixed. In order to keep the computational effort significantly small the basis functions, used to express the interface degrees of freedom, thereby are based on a statically reduced system model. By way of illustration, some examples of sub-structuring methods based on CMS are provided.
In Computers and Structures 79 (2001) 209, Tran describes a method for reducing the number of interface co-ordinates in a component mode synthesis method. The method is based on the use of a truncated basis of interface modes instead of the constraint modes and is illustrated for both free and hybrid interface methods. The obtained basis thereby is determined based on a statically reduced system model.
In proceedings of IMAC 96 (1996) 204, Balmès describes a CMS sub-structuring method whereby solutions for a complex system are obtained in a reduced sub-space corresponding to a reduction basis described by reduced DOFs. The difficulty is to choose the reduction basis such that for the qualities of interest similar results are obtained as for full models. Balmès uses interface displacement continuity conditions as generalised kinematic boundary conditions and interface force equilibrium conditions for model reduction through static condensation.
In the American Institute of Aeronautics and Astronautics Journal 39 (2001) 1182, Castanier et al. describe a technique for reducing the size of a model generated by a Craig-Bampton method. The method is based on performing an eigenanalysis on the constraint-mode partitions of the mass and stiffness matrices that correspond to the Craig-Bampton constraint modes. The method seems especially suited to predict power flow in complex structures.
Although the above mentioned papers have reported on the use of some sort of component modes as interface basis functions, further optimisation of the efficiency of the sub-structuring methods used for optimisation of designs still is needed, allowing efficient sub-structuring of (more) complex systems.